The type of mutation you are referring to is called a silent mutation. There are three things that can happen to an organism when a DNA mutation takes place: 1) something beneficial can happen, 2) something disastrous can happen, often resulting in the organism's demise, and 3) nothing can happen, the organism remains unchanged. Of the three, it is often the third one that takes place.
The reason nothing takes place has to do with the number of available codon combinations for that specific protein. If the point mutation changes one of the three bases to another base, and that codon combination codes for the same protein as the old codon combination, nothing happens to the protein. The end result is "business as usual", meaning the same protein is being constructed, just by a different codon that happens to do the same thing.
This is synonomous to building a road. There are many different companies that can build roads. If you start the road building project with one company, then decide you want to switch to another company, for what-ever reason, there are other companies to choose from which will provide the same end result.